What do you mean by kinetic energy derive an expression for the kinetic energy of an object of mass m moving with velocity v?

Let the initial velocity of the object be u. Let an external force be applied on it so that it gets displaced by distance s and its velocity becomes v. In this scenario, the kinetic energy of the moving body is equal to the work that was required to change its velocity from u to v.

Thus, we have the velocity−position relation as:

v2 = u2 + 2as

or

`s = (ν^2 - u^2)/(2a)`   ....(i)

Where, a is the acceleration of the body during the change in its velocity

Now, the work done on the body by the external force is given by:

W = F × s

F = ma …(ii)

From equations (i) and (ii), we obtain :
`W = ma xx ((ν^2 - u^2)/(2a)) = 1/2 m(ν^2 - u^2)`

If the body was initially at rest (i.e., u =0) , then : `W = 1/2mν^2 `Since kinetic energy is equal to the work done on the body to change its velocity from 0 to v, we obtain:

Kinetic energy , `E_k = 1/2mν^2`